Implications

We have the following indirect implication of form equivalence classes:

68 \(\Rightarrow\) 146
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
68 \(\Rightarrow\) 62 Subgroups of a free group and the axiom of choice, Howard, P. 1985, J. Symbolic Logic
62 \(\Rightarrow\) 146 The axiom of choice in topology, Brunner, N. 1983d, Notre Dame J. Formal Logic
note-26

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
68:

Nielsen-Schreier Theorem: Every subgroup of a free group is free.  Jech [1973b], p 12.

62:

\(C(\infty,< \aleph_{0})\):  Every set of non-empty finite  sets  has  a choice function.

146:

\(A(F,A1)\): For every \(T_2\) topological space \((X,T)\), if \(X\) is a continuous finite to one image of an A1 space then \((X,T)\) is  an A1 space. (\((X,T)\) is A1 means if \(U \subseteq  T\) covers \(X\) then \(\exists f : X\rightarrow U\) such that \((\forall x\in X) (x\in f(x)).)\)

Comment:

Back